Stokes Solutions to Low Reynolds Number Flows P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Surface Forces tending to behave like Body Forces !?!?!

Low Re Uniform Flow Pas A Sphere Again, at r=R: and at r : For a separable solution, we look for a form:

Method of Separation of variables The problem yields readily to a product solution  (r, ) = f(r)g().

Model Solution in First Operator

Use of Second Operator

The Momentum Equation as ODE f(r) is governed by the equi-dimensional differential equation: whose solutions are of the form f(r) ∝ rn It is easy to verify that n = −1, 1, 2,4 so that

Evaluation of Constants The boundary conditions

Inside the parentheses, the first two terms together represent an inviscid flow past a sphere. The third term is called the Stokeslet, representing the viscous correction. The velocity components follow immediately

Stokes Flow

Properties of Stokes Solution This celebrated solution has several extraordinary properties: 1. The streamlines and velocities are entirely independent of the fluid viscosity. This is true of all creeping flows. 2. The streamlines possess perfect fore-and-aft symmetry. There is no wake. 3. The local velocity is everywhere retarded from its freestream value. 4. The effect of the sphere extends to enormous distances: At r = 10a, the velocities are still about 10 percent below their freestream values. Stream lines in stokes flow are displace away from the sphere.

Severe Displacement of Stream Lines!!!!

Pressure Field for an Immersed Sphere With Stokes Flow Equation

Evaluation of Pressure Field Integrating with respect to r from r=a to r∞, we get where p, is the uniform free stream pressure. The pressure deviation is proportional to  and antisymmetric. Positive at the front and negative at the rear of the sphere This creates a pressure drag on the sphere.

Shear Stress Field There is a surface shear stress which creates a drag force. The shear-stress distribution in the fluid is given by

Stokes Drag on Sphere The total drag is found by integrating pressure and shear around the surface: This is the famous sphere-drag formula of Stokes (1851). Consists of two-thirds viscous force and one-third pressure force. The formula is strictly valid only for Re << 1 but agrees with experiment up to about Re = 1.

Experimental Validation of Stokes Solution

Other Three-Dimensional Body Shapes In principle, a Stokes flow analysis is possible for any three-dimensional body shape, providing that one has the necessary analytical skill. A number of interesting shapes are discussed in the literature. Of particular interest is the drag of a circular disk: Disk normal to the freestream: Disk parallel to the freestream:

Selected Analytical Solutions to NS Equations Couette (wall-driven) steady flows Poiseuille (pressure-driven) steady duct flows Unsteady duct flows Unsteady flows with moving boundaries Duct flows with suction and injection Wind-driven (Ekman) flows Similarity solutions (rotating disk, stagnation flow, etc.)

COUETTE FLOWS These flows are named in honor of M. Couette (1890). He performed experiments on the flow between a fixed and moving concentric cylinder. Steady Flow between a Fixed and a Moving Plate. Axially Moving Concentric Cylinders. Flow between Rotating Concentric Cylinders.

Steady Flow between a Fixed and a Moving Plate In above Figure two infinite plates are 2h apart, and the upper plate moves at speed U relative to the lower. The pressure is assumed constant. These boundary conditions are independent of x or z ("infinite plates"); It follows that u = u(y) and NS equations reduce to

Continuity Equation: NS Equations: where continuity merely verifies our assumption that u = u(y) only. NS Equation can be integrated twice to obtain

The boundary conditions are no slip, u(-h) = 0 and u(+h) = U. c1= U/2h and c2 = U/2. Then the velocity distribution is

The shear stress at any point in the flow From the viscosity law: For the couette flow the shear stress is constant throughout the fluid. The strain rate-even a nonnewtonian fluid would maintain a linear velocity profile.